Multithreshold rules

Multithreshold rules are natural extensions of the (binary-valued) generalized threshold rule defined in equation 5.121. Using fc-state variables a G 0,1. A - 1 and using the j entry of the A threshold vector to define the j threshold at that site (= bij), the generalized multithreshold rule is defined as follows  

Although it is certainly not immediately obvious from the rule itself, it turns out that, just as is the case for generalized threshold rules, the only possible asymptotic states of finite symmetric muib -threshold rules are either fixed points or cycles of period two [golesQO]. Unlike their binary brethren, however, multi-threshold rules possess some intriguing additional properties. 

For example, consider a radius-r one-dimensional neighborhood, set k — 4, and let bij = bj for all j (i.e. assign the same set of thresholds bo,bi and 62 to each site i). The asymptotic behavior of this rule - in particular, the question whether 

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